Sweep–Quench–Sweep Protocol in Quantum Systems

• The sweep–quench–sweep protocol is a three-stage control technique that combines gradual parameter sweeps with an abrupt quench, enabling coherent bypassing of small spectral gaps.
• It uses controlled diabatic transitions and phase accumulation to induce Stückelberg interference, resulting in nonthermal mode occupations and distinctive Loschmidt echo signatures.
• In Rydberg atom experiments, the protocol achieves efficient target state preparation by populating a low-lying, scar-like subspace, greatly outperforming linear sweeps.

The sweep–quench–sweep protocol is a piecewise coherent control protocol for many-body quantum systems in which a control parameter is first swept toward a difficult spectral region, then abruptly quenched and held for a finite duration, and finally swept again to the target Hamiltonian. In the literature summarized here, the protocol appears in two closely related forms. In a one-dimensional generalized quantum Ising model, it provides a natural extension of sweep-through-criticality dynamics, where the quench interval accumulates mode-dependent phases and the second passage generates Stückelberg interference, nonthermal mode occupations, and Loschmidt-echo structure (0907.3206). In programmable Rydberg atom arrays, it is implemented as a native-control algorithm that circumvents a superexponentially small first-order gap by coherently populating a small low-lying subspace before the bottleneck and then mapping that amplitude back to the target ground state during the second sweep (Lukin et al., 2024).

1. Protocol definition and control structure

The protocol has three stages. A first sweep moves the Hamiltonian from an initial parameter region toward a bottleneck in the spectrum. A quench then changes the control parameter suddenly and holds the Hamiltonian fixed for a waiting time. A second sweep resumes the ramp toward the final target. The shared objective is not strict adiabatic following throughout the evolution, but controlled use of diabaticity and phase accumulation.

In the generalized quantum Ising setting, the control parameter is $g$, with a linear sweep

$g(t) = g_i - (\Gamma/J)t,$

starting from the ground state $|\psi_0(g_i)\rangle$ of $H(g_i)$ and crossing the critical point at $g_c=0$. After the first sweep reaches $g_f$, the Hamiltonian is held fixed at $H_f \equiv H(g_f)$ during a post-sweep evolution. The explicit sweep–quench–sweep extension consists of a first passage across $g_c$ with rate $\Gamma_1$, a waiting interval under $H(g_f^1)$ for time $g(t) = g_i - (\Gamma/J)t,$0, and a second passage with rate $g(t) = g_i - (\Gamma/J)t,$1 (0907.3206).

In the Rydberg-array implementation, the control parameter is the detuning $g(t) = g_i - (\Gamma/J)t,$2 in

$g(t) = g_i - (\Gamma/J)t,$3

Here $g(t) = g_i - (\Gamma/J)t,$4 is typically held constant while $g(t) = g_i - (\Gamma/J)t,$5 is ramped linearly, jumped suddenly from $g(t) = g_i - (\Gamma/J)t,$6 to $g(t) = g_i - (\Gamma/J)t,$7, held for duration $g(t) = g_i - (\Gamma/J)t,$8, and then ramped again. The initial state is $g(t) = g_i - (\Gamma/J)t,$9, and the target is the ground state of the final Hamiltonian at large positive detuning, namely the maximum independent set (MIS) configuration of the chosen geometry (Lukin et al., 2024).

2. Spectral bottlenecks, adiabaticity, and diabatic control

The protocol is motivated by the failure of naive adiabatic evolution near small many-body gaps. The adiabatic condition is

$|\psi_0(g_i)\rangle$0

with a characteristic adiabatic time

$|\psi_0(g_i)\rangle$1

so the runtime becomes prohibitive when the minimum gap $|\psi_0(g_i)\rangle$2 is small. At isolated avoided crossings, the Landau–Zener intuition is

$|\psi_0(g_i)\rangle$3

which shows directly that very small gaps demand extremely slow sweeps (Lukin et al., 2024).

The two principal regimes studied in the cited works are distinct. In the generalized quantum Ising model, the central structure is a quantum critical point at $|\psi_0(g_i)\rangle$4. The model

$|\psi_0(g_i)\rangle$5

belongs to the $|\psi_0(g_i)\rangle$6D quantum Ising universality class with $|\psi_0(g_i)\rangle$7. Along the “thermal” direction, $|\psi_0(g_i)\rangle$8; along the “magnetic” direction, $|\psi_0(g_i)\rangle$9. The gap scales as

$H(g_i)$0

A slow sweep creates excitations with Kibble–Zurek scaling

$H(g_i)$1

so for $H(g_i)$2 and $H(g_i)$3 one obtains $H(g_i)$4 in the integrable transverse-field Ising direction and $H(g_i)$5 when the longitudinal component is present (0907.3206).

In the Rydberg doublet-chain setting, the bottleneck is instead a first-order transition between macroscopically distinct configurations. Including realistic van der Waals tails produces a zigzag state at small positive detuning and the MIS state at larger positive detuning, with a minimum gap that scales as

$H(g_i)$6

The two phases are separated by an extensively large Hamming distance,

$H(g_i)$7

which makes conventional adiabatic preparation ineffective at moderate runtime (Lukin et al., 2024).

3. Coherent mechanism in the generalized quantum Ising model

For the integrable case $H(g_i)$8, the post-sweep state after passage through the critical point factorizes into independent Bogoliubov modes,

$H(g_i)$9

with Landau–Zener excitation probabilities

$g_c=0$0

These occupations are strongly nonthermal, overpopulating low-$g_c=0$1 modes relative to any Boltzmann distribution tied to the final Hamiltonian. As a result, no single effective temperature characterizes observables in the post-sweep state (0907.3206).

The sweep–quench–sweep extension turns this critical-passage dynamics into a coherent interferometer. After the first sweep, waiting under $g_c=0$2 for time $g_c=0$3 accumulates a phase

$g_c=0$4

The second sweep then constitutes a second Landau–Zener traversal. In the integrable limit, the expected modification is mode-wise Stückelberg interference. For a symmetric double passage with $g_c=0$5,

$g_c=0$6

with $g_c=0$7. The consequence is that defect production can be either suppressed or enhanced by tuning $g_c=0$8 and the intermediate value $g_c=0$9, since these determine $g_f$0 (0907.3206).

This framework also fixes how a second sweep modifies later observables. The entanglement-growth slope after the second sweep is modulated through the altered mode occupations $g_f$1, and the Loschmidt echo can exhibit revived cusps at times determined by the new mode satisfying $g_f$2. This suggests an interferometric reading of the protocol: the wait interval stores coherent phase information, while the second passage converts it into observable changes in excitation density, entanglement production, and dynamical nonanalyticities.

4. Rydberg-array realization as a shortcut to adiabaticity

The experimental realization uses QuEra’s Aquila programmable neutral-atom simulator, with $g_f$3 atoms in optical tweezers driven globally between $g_f$4 and a Rydberg state $g_f$5. The typical Rabi rate is $g_f$6, so $g_f$7. The platform supports a quasi-$g_f$8D “Rydberg doublet chain” and a $g_f$9D array with embedded doublets. In the quasi-$H_f \equiv H(g_f)$0D geometry with side length $H_f \equiv H(g_f)$1, the interaction hierarchy is $H_f \equiv H(g_f)$2, $H_f \equiv H(g_f)$3, and the next-nearest-neighbor couplings satisfy $H_f \equiv H(g_f)$4, favoring zigzag order at small positive detuning (Lukin et al., 2024).

The piecewise protocol is explicit. The first sweep ramps $H_f \equiv H(g_f)$5 linearly from a large negative value up to $H_f \equiv H(g_f)$6 at constant $H_f \equiv H(g_f)$7. The quench then instantaneously changes the detuning from $H_f \equiv H(g_f)$8 to $H_f \equiv H(g_f)$9 and holds for duration $g_c$0. The second sweep resumes the linear ramp from $g_c$1 to large positive detuning. In the quasi-$g_c$2D experiments, the reported parameters that performed best were

$g_c$3

with

$g_c$4

and this optimal quench duration was nearly independent of system size across $g_c$5. In the $g_c$6D $g_c$7 array, the reported parameters were $g_c$8, $g_c$9, and $\Gamma_1$0 (Lukin et al., 2024).

The quench is sudden on the scale of the small gap, since $\Gamma_1$1 is trivially satisfied when $\Gamma_1$2 is superexponentially small. Its physical role is to inject coherent amplitude into special low-lying excited states before the first-order transition. During the resumed sweep, these states diabatically map back toward the true ground state beyond the tiny avoided crossing. The work identifies this low-lying sector as a small nonergodic, scar-like subspace: MIS probability shows revivals as a function of $\Gamma_1$3, the revival period depends nearly linearly on $\Gamma_1$4, and tensor-network simulations show that the dynamics remains in a low-entanglement corner of Hilbert space, with a bond dimension $\Gamma_1$5 sufficient to capture the dynamics with high fidelity (Lukin et al., 2024).

5. Dynamical observables and diagnostic signatures

The generalized quantum Ising analysis supplies a detailed set of observables for diagnosing sweep–quench–sweep dynamics. The excess energy above the ground state at the end of a sweep,

$\Gamma_1$6

scales as

$\Gamma_1$7

and therefore inherits the same $\Gamma_1$8 exponent as the excitation density when $\Gamma_1$9 is $H(g_f^1)$0-independent. In the integrable transverse-field Ising case, this gives $H(g_f^1)$1; in the longitudinal direction, $H(g_f^1)$2 (0907.3206).

Entanglement entropy exhibits two scaling regimes. Immediately after the sweep,

$H(g_f^1)$3

For the transverse-field Ising case with $H(g_f^1)$4 and $H(g_f^1)$5,

$H(g_f^1)$6

while for $H(g_f^1)$7 with $H(g_f^1)$8,

$H(g_f^1)$9

During subsequent evolution under the final Hamiltonian,

$g(t) = g_i - (\Gamma/J)t,$00

with oscillations set by the post-sweep gap scale. In weakly nonintegrable cases, the early-time linear growth persists only up to a timescale set by quasiparticle interaction and scattering, after which entanglement growth accelerates (0907.3206).

The Loschmidt amplitude and echo are

$g(t) = g_i - (\Gamma/J)t,$01

with thermodynamic-length scaling

$g(t) = g_i - (\Gamma/J)t,$02

In the integrable case,

$g(t) = g_i - (\Gamma/J)t,$03

At long times, $g(t) = g_i - (\Gamma/J)t,$04 algebraically, with

$g(t) = g_i - (\Gamma/J)t,$05

and the oscillation amplitude decays as $g(t) = g_i - (\Gamma/J)t,$06 rather than exponentially. Cusplike singularities occur when the special mode

$g(t) = g_i - (\Gamma/J)t,$07

satisfies $g(t) = g_i - (\Gamma/J)t,$08 and

$g(t) = g_i - (\Gamma/J)t,$09

Nonintegrable perturbations round these cusps and produce faster damping (0907.3206).

In the Rydberg experiments, the principal observable is the target-state success probability $g(t) = g_i - (\Gamma/J)t,$10. Baseline linear sweeps with $g(t) = g_i - (\Gamma/J)t,$11 show rapid decay of $g(t) = g_i - (\Gamma/J)t,$12 with system size, while the zigzag probability decays much more slowly. Under sweep–quench–sweep, $g(t) = g_i - (\Gamma/J)t,$13 exhibits pronounced revivals versus $g(t) = g_i - (\Gamma/J)t,$14, and the first revival provides the best tradeoff of speed and probability. For the longest quasi-$g(t) = g_i - (\Gamma/J)t,$15D chain measured, the optimal protocol boosts $g(t) = g_i - (\Gamma/J)t,$16 by nearly two orders of magnitude compared to a matched linear sweep; in the $g(t) = g_i - (\Gamma/J)t,$17D $g(t) = g_i - (\Gamma/J)t,$18 system, the improvement is about two orders of magnitude over the linear sweep as well (Lukin et al., 2024).

6. Integrability, robustness, and scope of applicability

A central distinction is between integrable and nonintegrable implementations. In the transverse-field Ising limit $g(t) = g_i - (\Gamma/J)t,$19, the dynamics reduces to independent $g(t) = g_i - (\Gamma/J)t,$20-modes, the Landau–Zener probabilities determine all occupations, the post-sweep state remains nonthermal, and the Loschmidt echo shows sharp cusps with algebraic $g(t) = g_i - (\Gamma/J)t,$21 damping. For $g(t) = g_i - (\Gamma/J)t,$22, or with perturbations such as a longitudinal field $g(t) = g_i - (\Gamma/J)t,$23, next-nearest-neighbor Ising coupling $g(t) = g_i - (\Gamma/J)t,$24, or a transverse interaction $g(t) = g_i - (\Gamma/J)t,$25, scattering between modes invalidates the independent-mode picture. The consequences are rounded minima in the Loschmidt echo, accelerated entanglement growth, and faster decay of coherent oscillations (0907.3206).

In the Rydberg implementation, robustness is empirical rather than symmetry-protected. The high-performance region in $g(t) = g_i - (\Gamma/J)t,$26 is broad, the optimal $g(t) = g_i - (\Gamma/J)t,$27 is approximately independent of $g(t) = g_i - (\Gamma/J)t,$28, and its shift with $g(t) = g_i - (\Gamma/J)t,$29 is roughly linear. The leading experimental imperfections are state-readout errors, with per-site misclassification probabilities $g(t) = g_i - (\Gamma/J)t,$30 and $g(t) = g_i - (\Gamma/J)t,$31, together with a control-response delay of about $g(t) = g_i - (\Gamma/J)t,$32. The reported analysis treats these effects explicitly. Failure modes are also specific: if $g(t) = g_i - (\Gamma/J)t,$33 is too close to or beyond the first-order transition, the quench can miss the desired low-lying subspace; if $g(t) = g_i - (\Gamma/J)t,$34 is too small or too large, the protocol either fails to generate strong revivals or excites unwanted states (Lukin et al., 2024).

The combined body of work places sweep–quench–sweep protocols in two complementary roles. First, they are a control method for reducing the impact of adiabatic bottlenecks, either by exploiting Stückelberg interference near a critical passage or by coherently accessing a scar-like low-lying manifold near a first-order transition. Second, they are a diagnostic of coherence and integrability: sharp cusps, slow algebraic damping, and persistent interference indicate integrable or weakly interacting dynamics, whereas cusp rounding, enhanced dephasing, and rapid entanglement growth signal nonintegrable scattering (0907.3206). A plausible implication is that the same protocol family can function both as a shortcut to adiabaticity and as a spectroscopic probe of nonergodic structure in many-body Hilbert space (Lukin et al., 2024).